Cut 1 lime into 4 to 6 wedges;
set aside for serving. Juice as many of the remaining limes as needed to
measure 1/4 cup.

Season the chicken breasts with
the 1 tsp. salt and 1/2 tsp. pepper. In a large saucepan over medium heat, warm
the olive oil. Add the chicken, skin side down, and cook until browned, about 5
minutes. Transfer the chicken to a plate.

Add the onion to the pan and sauté
until translucent, about 4 minutes. Add the garlic and chili and sauté until
fragrant, about 1 minute. Add the broth, water, lime juice and oregano, then
return the chicken to the pan.

Increase the heat to high and
bring the liquid to a boil, skimming off any foam that rises to the surface.
Reduce the heat to medium-low, cover partially and simmer until the chicken
shows no sign of pink when pierced with the tip of a sharp knife near the bone,
about 40 minutes.

Transfer the chicken to a carving
board and let cool slightly. Meanwhile, keep the soup at a simmer. Remove and
discard the skin and bones, and shred the chicken into bite-size pieces. Stir
the chicken into the soup.

Add the torn tortilla strips, if using. Adjust the
seasonings with salt and pepper.

Put the avocado,
cheese and lime wedges in separate small bowls. Ladle the soup into warmed
bowls and serve immediately. Pass the avocado, cheese and lime wedges
alongside.